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From: Lee Davidson on 7 Jun 2010 20:56
I'm addressing this post to people who like to solve basic problems
related to foundational questions.

Start with ZFC. Now let ZF0 consist of the restriction of ZFC to the
following axioms:

Extensionality
Pair-set
Unions
Subsets

Given this theory, we can define the concept of an ordinal, and prove
that the ordinals are linearly ordered, and, in fact, well ordered.
The definition I am thinking of is:

x is an ordinal iff x if epsilon-transitive, linearly ordered by
epsilon, and if for any subset y of x y contains an epsilon-least
element.

You can also define finite ordinals and prove induction (mathematical
or transfinite) and show how to do recursive definitions (on finite
ordinals or in the form of transfinite recursion). (Well, actually,
transfinite recursion probably needs the Axiom of Replacement, so just
say recursion on finite ordinals.)

Now, let ZF00 be ZF0 without the Axiom of Subsets. It is easy to show
that there are models for ZF00 in which the ordinals, as just defined,
are not linearly ordered.

Note however, that the null-set -- 0 -- is provable to be an element
of any ordinal.

Now, here's the problem: within ZF00 can you add a clause to the
ordinal definition such that you restrict attention to the usual
finite ordinals 0,1,2... and maybe transfinite ordinals, such that all
these more restricted "ordinals" -- call these "really really
ordinals" -- are linearly ordered? Definition has to admit 0,1,2....

I say you can't, but I haven't proven that.

BTW, if you add an appropriate formulation of the Axiom of Foundation
to ZF00, I believe you are able to do this. But I'd like to do this
without just the blanket assumption that all sets are well-founded.
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